Vol. 9, No. 9, 2015

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Classifying orders in the Sklyanin algebra

Daniel Rogalski, Susan J. Sierra and J. Toby Stafford

Vol. 9 (2015), No. 9, 2055–2119
Abstract

Let S denote the 3-dimensional Sklyanin algebra over an algebraically closed field k and assume that S is not a finite module over its centre. (This algebra corresponds to a generic noncommutative 2.) Let A = i0Ai be any connected graded k-algebra that is contained in and has the same quotient ring as a Veronese ring S(3n). Then we give a reasonably complete description of the structure of A. This is most satisfactory when A is a maximal order, in which case we prove, subject to a minor technical condition, that A is a noncommutative blowup of S(3n) at a (possibly noneffective) divisor on the associated elliptic curve E. It follows that A has surprisingly pleasant properties; for example, it is automatically noetherian, indeed strongly noetherian, and has a dualising complex.

Keywords
noncommutative projective geometry, noncommutative surfaces, Sklyanin algebras, noetherian graded rings, noncommutative blowing-up
Mathematical Subject Classification 2010
Primary: 14A22
Secondary: 16P40, 16W50, 16S38, 14H52, 16E65, 18E15
Milestones
Received: 8 April 2014
Revised: 10 March 2015
Accepted: 3 September 2015
Published: 4 November 2015
Authors
Daniel Rogalski
Department of Mathematics
University of California, San Diego
9500 Gilman Drive #0112
La Jolla, CA 92093-0112
United States
Susan J. Sierra
School of Mathematics
University of Edinburgh
James Clerk Maxwell Building
Peter Guthrie Tait Road
Edinburgh
EH9 3FD
United Kingdom
J. Toby Stafford
School of Mathematics
University of Manchester
Alan Turing Building
Oxford Road
Manchester
M13 9PL
United Kingdom